A Study of Transonic Flow and Airfoils. Presented by: Huiliang Lui 30 th April 2007
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1 A Study of Transonic Flow and Airfoils Presented by: Huiliang Lui 3 th April 7
2 Contents Background Aims Theory Conservation Laws Irrotational Flow Self-Similarity Characteristics Numerical Modeling Conclusion
3 Background Transonic regime Loosely defined region of flow around sonic speed (free stream velocities.8 M 1. ) Mixed regions of locally subsonic and supersonic flow. Unpredictable effect of shockwaves on the control surfaces
4 Aims Expand knowledge of aerodynamics and compressible flow Investigate the effects of transonic flow on airfoils Analyze behavior of pressure coefficient, C p
5 Theory: Conservation Equations Euler s Equations: Continuity ρ + tt ( ρv ) = Momentum DV ρ Dt = p Energy Dh ρ o Dt = ρ t
6 Theory: Irrotational Flow Vorticity V r Vorticity = for irrotational flow Define: velocity potential Φ such that Why? V r Φ Simplifies conservation equations into one governing equation [1]: Φ 1 a Φ a x x Φ Φ y xx Φ xy Φ + 1 a y Φ xφ a z Φ Φ yy xz Φ + 1 a Φ a y Φ z z Φ Φ yz zz =
7 Theory: Irrotational Flow Validity for transonic flow: Entropy across shock [1] s γ 1 3 ( M ) 1 1 R s 3( γ + 1) For transonic, M 1 1 Flow can be assumed as isentropic, and therefore irrotational!
8 Theory: Irrotational Flow Introduce perturbation velocity potential: Φ = V x + φ Governing equation simplifies to: (1 = M M ) φ xx + φ Note: RHS drops out for subsonic or supersonic flow, resulting in linearized PDE. yy φx ( γ + 1) V + φ φ xx zz
9 Theory: Self-Similarity Recall governing equation: (1 M φ φ φ ) xx + yy + zz = M ( + 1) γ φx V φ xx Introduce slenderness ratio τ = b / c
10 Theory: Self-Similarity Self-similar variables: x = x c Nondimensionalize: y = φ = yτ c cv φ 1 3 Transonic similarity equation: where K = transonic similarity parameter: τ 3 z = zτ c [ K ( γ + 1) φ ] φ + φ + φ = K x xx 1 M = 3 τ yy zz 1 3
11 Theory: Characteristics Recall governing equation (D): 1 1 = Φ Φ Φ Φ Φ + Φ Φ xy y x yy y xx x a a a From midterm: 1 1 = + + a v dx dy a uv dx dy a u
12 Theory: Characteristics Solving, dy dx = uv ± a u + v ( a u ) a subsonic sonic supersonic elliptic parabolic hyperbolic Characteristic Slopes? Interpretation: The Mach Cone
13 Theory: Characteristics From NASAexplore s website:
14 Numerical Modeling Pressure coefficient po p CP = 1 ρv National Advisory Committee for Aeronautics (NACA) Data for foils Panel Methods
15 Panel Methods Basic principle: Superposition Boundary element method: Panels Sources/Sinks (simple solution) Vortices The Kutta Condition: Pressure above and below trailing edge must be equal
16 Strategy Attempt Vortex Panel method for three symmetric airfoils for linearized full potential equation (FPE) Help: ME163 website Extend to transonic modeling
17 Results: Symmetric Airfoils 8 NACA-1 Foil 8 NACA-15 Foil 6 6 y-c coordinate in % Airfoil Chord y-coordinate in % Airfoil Chord % Airfoil Chord % Airfoil Chord 1 NACA-18 Foil 8 y-coordinate in % Airfoil Chord % Airfoil Chord
18 Results: Linearized FPE 1.5 lower part upper part 1.5 lower part upper part C p -1 C p x-position x-position 1.5 lower part upper part -.5 C p x-position
19 Transonic Modeling Numerical solution is exponentially harder to obtain because of nonlinearity Make use of characteristics Further steps needed: Grid Generation: Solve FPE at nodes Discretization of the PDE Iterative solution
20 Sample Grid for NACA-1 1 x field panels (from GA Tech)
21 Results 1. Mach Number =.8 1. Mach Number = Mach Number = C p.6.4. C p.6.4. C p Mach Number = Mach Number = C p C p
22 Future Work Generate one case for nonlinear, transonic flow, and solve iteratively Validate with results from Oskam s article [5]: Transonic Panel Method for the Full Potential Equation Applied to Multicomponent Airfoils
23 Conclusion Better understanding of aerodynamics Application of mathematical methods for modeling Numerical modeling for nonlinear PDEs is significantly tougher than linearized PDEs Simplify PDEs whenever possible!
24 References 1.Anderson, J.D. Modern Compressible Flow. Houghton E.L. and Carpenter, P.W. Aerodynamics for Engineering Students 3. Ferrari, C. and Tricomi F.G. Transonic Aerodynamics 4. AE 393/493 Airfoil Design 5. Oskam, B. Transonic Panel Method for the Full Potential Equation Applied to Multicomponent Airfoils 6. ME163 Fall 6 Project Vortex Panel Method
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